A Stabilized Numerical Framework for Necrotic Tumor Growth via Coupled Boundary Integral and Obstacle Solvers

TL;DR

The paper tackles robust simulation of tumor growth governed by the incompressible Porous Media/Hele-Shaw limit, focusing on the challenging inner necrotic boundary defined by an obstacle problem. It develops a stabilized, two-boundary framework that couples nutrient diffusion, pressure fields, and domain evolution via a predictor–corrector strategy integrated with boundary-integral and kernel-free boundary-integral solvers on Cartesian grids, and augmented-Lagrangian obstacle solvers. A formal convergence analysis is provided for the single-interface (fully viable) case, while the necrotic-core model is demonstrated to be stable and capable of capturing nucleation and complex topology changes through numerical experiments. The results show second-order accuracy for potential solves and first-order convergence for boundary velocity in the non-necrotic regime, and robust multi-interface performance with accurate necrotic-core evolution in the presence of topological transitions. This framework offers a scalable, mesh-free approach for two-dimensional tumor simulations and lays the groundwork for extensions to more complex nutrient models and three-dimensional settings.

Abstract

We present a robust computational framework for Hele-Shaw tumor growth with necrotic cores, a problem identified as the incompressible limit of the Porous Media Equation. Simulating this system presents a fundamental challenge: while the outer boundary evolves via advection, the inner necrotic interface is defined by an obstacle problem and lacks an explicit advection structure, causing standard schemes to fail. To address this, we introduce a stabilized predictor-corrector strategy that iteratively resolves the bidirectional coupling between the nutrient-pressure fields and the domain geometry, ensuring robust time-stepping for both the advection-driven outer surface and the obstacle-defined necrotic core. We establish rigorous convergence theory for the single-interface case and demonstrate the method's robustness in capturing the topological transition of necrotic core nucleation and complex geometric evolution.

Figures (15)

• Figure 1: Schematic plot for the free boundary model.
• Figure 2: Predictor--corrector strategy for updating the inner and outer boundaries.
• Figure 3: Time evolution of the numerical solution for the radially symmetric test case. The computations are performed with $I=J=256$ and $\Delta t=0.01$.
• Figure 4: Evolution of the tumor boundary with $\lambda=1$.
• Figure 5: Evolution of the tumor boundary with $\lambda=50$.

Theorems & Definitions (21)

• Remark 1
• Remark 2
• Lemma 3.1: Second-order accuracy of the BI&KFBI solver for the Poisson problem
• proof : Proof sketch
• Lemma 3.2: Second-order accuracy with perturbed source term
• proof : Proof sketch
• Proposition 3.1: Gradient and velocity error
• Remark 3
• Theorem 3.2
• proof